71 Kronecker’s idea of arithmetization of mathematics Yoshiaki UENO 1 According to the 19 century German mathematician, Leopold Kronecker, the whole body of mathematics should be constructed rigidly on the basis of intuition of nat- ural numbers. His standpoint is historically called ‘intuitionism’. In this article, we examine Kronecker’s assertion about the ascendancy of natural numbers over other mathematical entities as the starting building block of mathematics from the per- spective of cognitive study of mathematics. 1 Introduction Rikitaro Fujisawa was one of the first Japanese scholars who studied European-style mathemat- ics. He is also famous for authoring the first series of national arithmetic and algebra textbooks for elementary and middle schools compiled by the Japanese restoration government. In the pre- vious paper [1], we considered how Fujisawa’s construction of his pedagogic thoughts—so-called ‘enumerationism’—were affected by Leopold Kronecker’s lecture in 1887. Although enumera- tionism textbooks were shortly replaced by the ‘green-cover series’, his thoughts and methods have had a great influence on the Japanese tradition of mathematics education until today. In the present article we look more closely at Kronecker’s own number theory textbook and examine his thoughts on the role of natural numbers in the foundations of mathematics. 2 Background: The discretization program Perhaps, “discretization” is one of the keywords to understand the history of 19th century European mathematics. G. Lakoff and R. N´u˜neswrote [2, p.262]: In late 19th century Europe, after the success of analytic geometry and calculus, mathematics had gained an important stature as being the discipline that defined the highest form of reason, with precise, rigorous, and indisputable methods of proof. Anything not formalizable was seen as “vague”, “intuitive” (as opposed to “rigor- ous”), and imprecise. At the same time, the foundation of mathematics was seen as crucial to preserving that stature for mathematics. This movement is later called ‘the discretization program’. Kronecker was a 19-century German mathematician, who insisted that the whole body of mathematics should be constructed rigidly on the basis of intuition of natural numbers, and denied the bold arguments of set theory. His standpoint is historically called ‘intuitionism’. In intuitionism, both truths and objects of mathematics do not stand independent of mind, but they 1 AssociateProfessor,GeneralEducationandResearchCenter,TokyoPolytechnicUniversity, [email protected] ei.ac.jp Received Sept. 10, 2003 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 26 No.1 (2003) 72 can be grabbed directly by mental activity. Later, Brouwer defined intuitionism more strictly. Brouwer did not allow the use of the law of the excluded middle, and he showed a keen conflict with Dedekind and Cantor. In what follows, we examine Kronecker’s assertion about the ascendancy of natural numbers over other mathematical entities as the starting building block of mathematics from the perspec- tive of cognitive study of mathematics. 3 Kronecker’s idea of arithmetization of mathematics In 1901, Kronecker published his lecture note on number theory [3]. He began this book by giving a brief history of number theory from Babylonian age to 19th century. Although his first concern in this book was number theory, he paid much attention on geometry and analysis in his introductory notes. At the first stage of his lecture, he mentioned that the numbers, especially the whole numbers, were the first mathematical entity of mankind. Gauss applied analytic methods to number theory and proved the very main theorem of number theory using analysis, and by doing so, Gauss raised a serious problem to the mathematical community in those days about the boundary between number theory and analysis. Kronecker, mentioning Gauss’s celebrated results, pointed out that many of the most fundamental constants of geometry and physics had, in their very basic definition, purely arithmetical properties. For example, Kronecker points out [3, p.4] that the most famous transcendental number ¼ can be defined by the Leibniz series ¼ 1 1 1 X1 (¡1)n = 1 ¡ + ¡ + ¢ ¢ ¢ = ; 4 3 5 7 2 n + 1 n=0 and, he mentions, that this exact (gerade in German) representation provides one of the most beautiful arithmetic properties of this vague (ungerade) irrational number, which stems from geometry. He writes: ::: in diesem Sinne ist wohl jenes bekannte Wort: ,,numero impari deus gaudet“ zu verstehen. (::: in this sense, that famous phrase: “the god loves odd numbers” can be understood well [4].) Indeed, the terms in this series can be distinguished remarkably well arithmetically; the sign of each term varies according to whether the denominator leaves 1 or 3 as residue when divided by 4. Kronecker states that: Wir haben hier also eine Definition der Transcendenten ¼ von durchaus zahlenthe- oretischem Charakter. (We have here, therefore, a definition of the transcendental number ¼ of completely number-theoretical character.) There are many other formulae including the number ¼, and some of them have been used to calculate the digits of ¼, but Leibniz formula is perhaps one of the simplest and most beautiful equations, though it converges very slowly. Kronecker’s idea of arithmetization of mathematics 73 Kronecker raised another example, including a continued fraction, which is as simple as the above one. z2 z ¢ tan z = z2 1 ¡ z2 3¡ 2 5¡ z . .. The only difference is that it serves as an implicit representation of the Rudolf number when one puts z = ¼=4 in the both sides. Here again, we see a simple arithmetic sequence of natural numbers [5]. So, what do these examples teach us? These examples are especially beautiful, but Kronecker argues that what we can learn from these examples can be applied universally to all definitions of analysis, i.e., all the definitions appearing in the field of analysis can be reduced to the whole numbers and their properties, and that the whole domain of this branch of mathematics can be explained basically from arithmetic. Therefore, by doing so, we can free analysis from its source domain—geometry. This meant for Kronecker that the boundary of arithmetic could be broadened substantially, and that arithmetic could serve as the foundation of all areas of mathematics, including geometry and physics. Here, Kronecker proposes a new word: “general arithmetic” (allgemeine Arithmetik), which includes algebra and analysis. According to Cantor’s set theory, there are much more transcendental numbers than algebraic numbers. On the real line, algebraic numbers are rare compared with other real numbers. But in ordinary mathematics, transcendental numbers we usually treat are not general transcendental numbers, but very special transcendental numbers like ¼, e, and ¼2, etc. Also among rational numbers, we put special emphasis on numbers like 0, 1, and i in mathematics. After all, these numbers are not just numbers like any other numbers. Unlike general rational and irrational numbers, these numbers have conceptual meanings, which is not just the numerical values. It depends on the idea of what mathematics is, but we can safely say that, for Kronecker, the object of mathematics was only those numbers which can be defined through specific procedures. From the standpoint of concept analysis of mathematics, which is a new branch of cognitive science of today, numbers like ¼, e, 0, 1, and i have conceptual meaning in a system of common, important, nonmathematical concepts [2, p.450]. 4 Ordinal numbers and cardinal numbers Kronecker’s lecture on number theory had some remarkable features. We point out here that there was a strict distinction between cardinal numbers and ordinal numbers in Kronecker’s textbook. The distinction between cardinal numbers and ordinal numbers is generally not mentioned in today’s number theory textbooks. The reason is that in today’s mathematical context, these concepts are just concerned about the usage of numbers in everyday life, and the distinction between them is unified, at the level of mathematics, into the single concept of ‘natural numbers’, which is the object of mathematics. These concepts of cardinals and ordinals are treated today, not in a course of mathematics, but rather in a course of linguistics. However, for Kronecker and his contemporary mathematicians, the circumstances were quite 74 ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 26 No.1 (2003) different. After the introductory three lectures of Kronecker’s treatment on number theory comes the first part. This part, entitled “Decomposability and congruence in the realm of numbers”, is about multiples, divisors, and congruence relations of natural numbers—a standard exposition of elementary number theory. At the very start of this part, Kronecker defines numbers and their operations (addition and multiplication). Let us look at the contents of the fourth lecture i.e. the starting lecture of this part: systematic arithmetic, the concept of numbers, the ordinal numbers, the cardinal numbers, the concept of size, addition, commutativity of the addition, the multipli- cation, commutativity of the factors of a product. We see from this list that the concept of numbers is immediately followed by the concept of ordinal numbers, and the concept of cardinal numbers comes afterwards. Why? Here we see strict difference between Kronecker’s concept of numbers and that of the set theorists. As a matter of fact, Kronecker did not define numbers. He just stated that the act of counting is a natural endowment of human kind. For him numbers are just the symbols 1; 2; 3; 4; 5;::: standing in a line. It was not important what kind of symbols were used; the important point here was the fact that we human beings can immediately understand the structure of the system constructed by these symbols—that these symbols are distinct from each other, that a linear ordering is given a priori, with each symbol followed by another symbol, and that the sequence starts with a unique symbol and continues indefinitely.